Modelling the control of tsetse and African
trypanosomiasis through application of insecticides on
cattle in Southeastern Uganda
Supervisor: Prof. John Hargrove
Co-supervisors: Prof. Joseph Y.T. Mugisha & Prof. Charles Waiswa by
Damian Kajunguri
Dissertation presented at the University of Stellenbosch for the degree of
Doctor of Philosophy
Department of Mathematical Sciences University of Stellenbosch
I, the undersigned, hereby declare that the work contained in this thesis is my own original work and has not previously, in its entirety or in part, been submitted at any
university for a degree.
- - -
-Damian Kajunguri Date
Copyright ©2013 Stellenbosch University
Abstract
In Uganda, cattle are an important reservoir of Trypanosoma brucei rhodesiense, a parasite that causes human African trypanosomiasis or sleeping sickness. We developed mathemat-ical models to examine the transmission of T. b. rhodesiense by tsetse vector species, Glossina fuscipes fuscipes in a host population that consists of humans, domestic and wild mammals, and reptiles. The models were developed and analysed based on the situation in Tororo district in Southeastern Uganda, where sleeping sickness is endemic and which has a cattle and human population of 40, 000 and 500, 000, respectively. Assuming populations of cattle and humans only, the impact of mass chemoprophylaxis and vector control through insecticide-treated cattle (ITC) is evaluated. Keeping 12% or 82% of the cattle population on insecticides that have an insecticidal killing effect of 100% at all times or trypanocides that have 100% efficacy, respectively, can lead to the control of T. b. rhodesiense in both humans and cattle. Optimal control of T. b. rhodesiense is shown to be achieved through ITC alone or a combination of chemoprophylaxis and ITC, the former being the cheapest control strategy. Allowing for the waning effect of insecticides and including wildhosts, T. b. rhodesiense control can be achieved by keeping 21% or 27% of the cattle popula-tion on insecticides through whole-body or restricted applicapopula-tion, respectively. Restricting the treatment of insecticides to adult cattle only would require 24% or 33% of the adult cattle population to be kept on insecticides through whole-body or restricted application, respectively, to control T. b. rhodesiense. A cost-effectiveness and benefit-cost analysis of using ITC to control T. b. rhodesiense show that restricted application of insecticides is a cheaper and more beneficial strategy compared to whole-body treatment. The results of the study show that the restricted application of insecticides on cattle provides a cheap, safe and farmer-based strategy for controlling tsetse and trypanosomiasis.
In Uganda is beeste ’n belangrike reservoir van Trypanosoma brucei rhodesiense, ’n parasiet wat tripanosomiase of slaapsiekte in mense veroorsaak. Ons het wiskundige modelle on-twikkel wat die oordrag van T. b. Rhodesiense deur tesetse vektor spesies, Glossina fuscipes fuscipes in ’n draer populasie wat bestaan uit mense, mak en wilde diere en reptiele, on-dersoek. Die modelle was ontwikkel en geanaliseer gebaseer op die oordrag situasie in die Tororo distrik in Suidoostelike Uganda, ’n gebied waar slaapsiekte endemies is en wat ’n populasie van 40, 000 beeste en 500, 000 mense het. Die impak van massa chemoprofilakse en vektor beheer deur insekdoder-behandelde beeste is gevalueer onder die aanname van bees en mens populasies alleenlik. Beheer oor T. b. Rhodesiense in beide mense en beeste kan verkry word deur of 12% van die bees populasie te behandel met ’n insekdoder wat 100% effektief is ten alle tye of 82% van die bees populasie te behandel met tripanosiedes wat 100% effektief is. Daar is aangetoon dat optimale beheer van T. b. Rhodesiense bereik kan word deur die gebruik van insekdoders alleenlik of ’n kombinasie van insekdo-ders en chemoprofilakse, hoewel eersgenoemde die goedkoopste strategie is. Wanneer die kwynende effek van insekdoders asook wilde diere as draers in ag geneem word, kan T. b. Rhodesiense beheer verkry word deur 21% van beeste se hele liggaam met insekdoders te behandel of 27% gedeeltelik te behandel. As slegs volwasse beeste met insekdoders behandel word, moet 24% se hele liggaam of 33% gedeeltelik behandel word vir beheer van T. b. Rhodesiense. ’n Koste-effektiwiteit en voordeel-koste analise van insekdoders as beheermaatstaf vir T. b. Rhodesiense toon aan dat gedeeltelike behandeling van die bees se liggaam die goedkoper en meer voordelige strategie is in vergelyking met behandeling van die hele liggaam. Die resultate van die studie wys dat gedeeltelike behandeling van beeste met insekdoders ’n goedkoop, veilige en landbouer-gebaseerde strategie is om tsetse en tripanosomiase te beheer.
Acknowledgments
I would like to thank my supervisor, John Hargrove for his guidance and advice towards the success of this project. I would also like to thank him for leading me into this field of Mathematical Epidemiology which I started when I was still at AIMS in 2007 from where I did an essay in the same area with him as my supervisor.
Special thanks goes to ICONZ for funding my PhD course and SACEMA for providing all the support and making my stay at the research centre during the three years of my study a memorable one. Without your support, this work would have been impossible.
I am also indebted to my co-supervisors, JYT Mugisha and Charles Waiswa for their generous effort towards the completion of this project. Thanks for your kindness and patience during this entire project.
I want to thank Dr. Rachid Ouifki for his assistance. Throughout this project he made valuable contributions especially in the model design and mathematical analysis. His assistance is greatly appreciated.
I will not fail to acknowledge Susan Welburn, Paul Coleman, Alex Shaw and Dennis Muhanguzi for their academic contributions towards this project. I am also grateful to Gavin Hitchcock for his assistance in editing this thesis. To the SACEMA family, I say thank you for working with me at the centre and providing good company.
Finally, my sincere appreciation goes to my family and friends at home, who instilled in me the strength and confidence to continue in the way of success, to make my passion and dreams come true.
1 Introduction 1
1.1 Background . . . 1
1.2 Epidemiology of African trypanosomiasis in Africa . . . 1
1.3 Epidemiology of African trypanosomiasis in Uganda . . . 4
1.4 Importance of domestic animals and wildlife in the transmission of African trypanosomiasis . . . 6
1.5 Parasite lifecycle and clinical aspects of infection . . . 7
1.6 Control of African trypanosomiasis . . . 8
1.7 Cost-effectiveness analysis of disease control strategies . . . 10
1.8 Benefit-cost analysis of disease control . . . 11
1.9 Integrated Control of Neglected Zoonotic diseases (ICONZ) project . . . . 11
1.10 Motivation . . . 12
1.11 Objectives of the study . . . 13
1.12 Outline of this work . . . 13
1.13 Publications . . . 14
2 Literature review 16
Contents ii
2.1 Modelling the transmission and control of trypanosomiasis . . . 16
2.2 Comparative analysis of the costs, benefits and effectiveness of tsetse and trypanosomiasis control . . . 19
2.3 Conclusion . . . 20
3 Modelling the control of T. b. rhodesiense through mass chemoprophy-laxis and insecticide-treated cattle 22 3.1 Introduction . . . 22
3.2 Model development and analysis . . . 22
3.2.1 Estimation of the tsetse mortality in the presence of insecticide-treated cattle . . . 25
3.2.2 Effect of mass chemoprophylaxis and ITC on R0 . . . 29
3.2.3 The optimal control problem to minimise the cost of controlling T. b. rhodesiense through chemoprophylaxis and ITC . . . 30
3.2.4 Sensitivity analysis . . . 31
3.3 Parameter estimation . . . 37
3.3.1 Demographic parameters . . . 37
3.3.2 Tsetse infection parameters . . . 38
3.3.3 Cattle infection parameters . . . 39
3.3.4 Human infection parameters . . . 39
3.4 Numerical results . . . 39
3.4.1 Estimation of minimum R0H if humans only are susceptible to infection 39 3.4.2 Prevention of an epidemic . . . 40 3.4.3 Optimal control of the disease through chemoprophylaxis and ITC . 46
3.5 Summary . . . 50
4 A multi-host model for the control of tsetse and T. b. rhodesiense through insecticide-treated cattle 52 4.1 Introduction . . . 52
4.2 Model development . . . 52
4.2.1 Including insecticide-treated cattle (ITC) in the model . . . 55
4.3 Mathematical analysis . . . 58
4.3.1 Analysis of the multi-host model in the absence of ITC . . . 58
4.3.2 Local stability of the disease-free equilibrium of the one-host T. b. rhodesiense transmission model in the absence of ITC . . . 61
4.3.3 The endemic steady states of the multi-host T. b. rhodesiense trans-mission model in the absence of ITC . . . 64
4.3.4 Mathematical analysis of the multi-host model in the presence of ITC 66 4.3.5 Sensitivity analysis of RT 03 to parameter values . . . 67
4.3.6 Cost-effectiveness analysis . . . 69
4.4 Numerical results . . . 70
4.4.1 Parameter estimation . . . 71
4.4.2 Importance of cattle and wildlife in the transmission of T. b. rhode-siense . . . 74
4.4.3 Application of insecticides to cattle to prevent the transmission . . 76
4.4.4 Effect of insecticide-treated cattle on the tsetse population . . . 80
4.4.5 Cost-effectiveness analysis of the ITC control programme . . . 80
Contents iv
5 Modelling the control of tsetse and T. b. rhodesiense by treating adult
cattle only with insecticides 85
5.1 Introduction . . . 85
5.2 Model development . . . 86
5.2.1 Basic reproduction number . . . 88
5.2.2 Benefit-cost analysis . . . 89
5.3 Numerical Results . . . 90
5.3.1 Prevention of the epidemic by treating adult cattle only with insec-ticides . . . 90
5.3.2 Effect of treating adult cattle only with insecticides on the tsetse population . . . 93
5.3.3 Benefit-cost analysis of the ITC control programme . . . 94
5.4 Summary . . . 95
6 Conclusions and recommendations 97 6.1 Limitations and future work . . . 99
Appendix 101
1.1 Distribution of human African trypanosomiasis in sub-Saharan Africa. The black line divides the areas in which Trypanosoma bruce gambiense prevails and in which Trypanosoma brucei rhodesiense predominates [11]. . . 2 1.2 Number of reported cases of African trypanosomiasis and population screened,
1939-2004. Data obtained from [55, 76] . . . 3 1.3 The map of Uganda highlighting areas of T. b. gambiense and T. b
rhode-siense transmission for the years 1985, 1995, 2000 and 2005 [26]. . . 4 1.4 Number of reported cases of African trypanosomiasis in Uganda, 1939-2004.
Data obtained from [55, 76] . . . 5 1.5 Number of reported cases of T. b. rhodesiense in Uganda compared with
all the cases reported in Africa, 1990-2009. Data obtained from [55, 76] . . 6 1.6 The life cycle of African trypanosomiasis. Reproduced from [11] . . . 7 3.1 Flow diagram of the compartmental model of T. b. rhodesiense in cattle,
human and tsetse populations. λC, λH, and λV are the forces of infection
for cattle, humans and tsetse vectors, respectively. . . 25 3.2 Sensitivity analysis of R0 with respect to φC for ǫ = 100%, 90%, 85%, 75%
and 65% (a), and the control parameters p and φC for ǫ = 100% (b). All
the other parameters are given in Table 3.5. . . 33
List of figures vi
3.3 The effect of mass chemoprophylaxis on the basic reproduction number for different values of ǫ and γC. The values of γC and ǫ were taken to be 0.0055
and 100% in (a) and (b), respectively. . . 41 3.4 Proportion of cattle kept on trypanocides for different values of φC. . . 41
3.5 Mass chemoprophylaxis. (a) incidence and (b) prevalence of T. b. rhode-siense in humans. Different proportions of cattle are taken to be kept on chemoprophylactic drugs that provide 100% protection to infection. The recovery rate for infectious cattle gC is taken to be g
′
C+ φC. . . 42
3.6 The effect of insecticide-treated cattle on the basic reproduction number. . 43 3.7 Insecticide-treated cattle. (a) incidence and (b) prevalence of T. b.
rhode-siense in humans for different proportions of cattle assumed to be kept on insecticides. . . 44 3.8 The effect of combining chemoprophylaxis and insecticide-treated cattle on
the basic reproduction number for ǫ = 100% (a), and different values of ǫ (b). 45 3.9 Insecticide-treated cattle and mass chemoprophylaxis. (a) incidence and (b)
prevalence of T. b. rhodesiense in humans. 2.0% of the cattle population is assumed to be kept on insecticides and 3.5% of cattle is treated with trypanocides that provide 100% protection to infection. The recovery rate for infectious cattle gC is taken to be gC′ + φC. . . 46
3.10 The critical proportions of cattle needed to be kept on insecticides and treated with trypanocides for R0 to be less than ρ for ρ = 1 (a) and ρ = 0.5
(b). The cost ratio (CR) refers to the ratio of the daily cost of keeping one cow on insecticides to the cost of treating one cow with trypanocides. ǫ = 1 in all cases and all the other parameters values are given in Table 3.5. . . . 47 3.11 Insecticide-treated cattle and mass chemoprophylaxis. Proportions of cattle
required to be treated with either insecticides or trypanocides for CR = 0.04 (a) and CR = 0.12 (b) . ǫ = 1 in both cases and all the other parameters values are given in Table 3.5. . . 48
3.12 Insecticide-treated cattle and mass chemoprophylaxis. Total cost for control-ling T. b. rhodesiense in humans and cattle against the basic reproduction number threshold (ρ) and the time required for controlling the disease in years. ǫ = 1 in all cases and all the other parameters values are given in Table 3.5. . . 48 4.1 Flow diagram of the compartmental model of T. b. rhodesiense in tsetse and
a multi-host population that includes treatment of cattle with insecticides. The cattle population that is treated with insecticides is enclosed in a dotted rectangle. λiand λV are the forces of infection for the host and tsetse vector
populations, respectively. . . 54 4.2 Tsetse mortality estimates for different proportions of cattle treated with
insecticides. . . 56 4.3 Simulation results for the T. b. rhodesiense incidence and prevalence in
humans in the absence of curative treatment of infectious humans and cattle, and ITC. . . 73 4.4 T. b. rhodesiense infection rates for cattle, human and wildlife host
popu-lations. . . 74 4.5 T. b. rhodesiense infection rate in wildlife with different proportions of
monitor lizards in the wildlife population . . . 75 4.6 T. b. rhodesiense incidence (a) and prevalence (b) in humans for different
proportions of monitor lizards in the wildlife population. . . 76 4.7 The effect of treating cattle with insecticides on the basic reproduction
num-ber, RT
03. (a) shows the plot of the basic reproduction number, RT03 with
respect to the proportion of cattle treated with insecticides, ψ for d = 0.04. (b) and (c) shows the critical proportions of cattle needed to be treated with insecticides for different values of d for whole body and restricted applica-tion, respectively. . . 77 4.8 Proportion of cattle kept on insecticides . . . 78
List of figures viii
4.9 Increased tsetse mortality through insecticide-treated cattle (ITC). T. b. rhodesiense incidence in humans for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . . 78 4.10 Increased tsetse mortality through insecticide-treated cattle (ITC). T. b.
rhodesiense prevalence in humans for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . . 79 4.11 Effect of increased tsetse mortality through insecticide-treated cattle on
the tsetse population for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . . 80 4.12 The cost of implementing the insecticide-treated cattle (ITC) control
pro-gramme over time for whole-body (a) and restricted application of insecti-cides (b) to cattle . . . 81 4.13 The cost-effectiveness ratio per infection avoided of the insecticide-treated
cattle (ITC) control programme over time for whole-body (a) and restricted application of insecticides (b) to cattle . . . 82 4.14 The cost-effectiveness ratio per DALY averted of the insecticide-treated
cat-tle (ITC) control programme over time for whole-body (a) and restricted application of icnsectcides (b) to cattle . . . 82
5.1 Flow diagram of the compartmental model of T. b. rhodesiense in tsetse, cattle, human and wildlife populations that includes treatment of adult cat-tle only with insecticides. The adult catcat-tle population that is treated with insecticides is enclosed in a dotted rectangle. λi and λV are the forces of
5.2 The effect of treating adult cattle only with insecticides on the basic repro-duction number, RT
A. (a) shows the plot of the basic reproduction number,
RT
A with respect to the proportion of adult cattle treated with insecticides,
ψA for d = 0.04. (b) and (c) shows the critical proportions of adult cattle
needed to be treated with insecticides for different values of d for whole body and restricted application, respectively. . . 91 5.3 Proportion of adult cattle kept on insecticides . . . 92 5.4 Increased tsetse mortality through treatment of adult cattle only with
insec-ticides. T. b. rhodesiense incidence in humans for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . 93 5.5 Increased tsetse mortality through treatment of adult cattle only with
in-secticides. T. b. rhodesiense prevalence in humans for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . . 93 5.6 Effect of increased tsetse mortality through treatment of adult cattle with
insecticides on the tsetse population for both strategies; whole-body treat-ment (a) and restricted application of insecticides to cattle (b). . . 94 5.7 The benefit-cost ratio of the ITC control programme through treatment of
adult cattle only with insecticides for both strategies; whole-body treatment (a) and restricted application of insecticides to cattle (b). . . 95
List of Tables
3.1 The model variables . . . 23 3.2 Definitions of the parameters used in the model . . . 24 3.3 Sensitivity indices of R0. All parameters were fixed to the values given in
Table 3.5, φC, ǫ and p were taken to be 0.02, 0.5 and 0.05, respectively. . . 32
3.4 Sensitivity indices of the endemic equilibrium with respect to the control parameters considered in the model. All parameters were fixed to the values given in Table 3.5, φC, ǫ and p were taken to be 0.05, 0.8 and 0, respectively.
The tsetse recruitment and mortality rates are evaluated at p = 0 and taken to be ΛV = ΛV(0) = 1, 235 and µV = µV(0) = 0.03, respectively, where µV
is the tsetse natural mortality. . . 36 3.5 Model parameter value estimates. All time periods are given in days and all
rates are instantaneous per capita rates per day. . . 38
4.1 Definitions of the parameters used in the model . . . 53 4.2 Sensitivity indices of RT
03 of the 3-host T. b. rhodesiense model with and
without insecticide-treated cattle. All parameters were fixed to the values given in Table 4.3, and ψ was taken to be 0.02. . . 68 4.3 Numerical values for the parameters of the three-host model . . . 72
Introduction
1.1
Background
Human African trypanosomiasis or sleeping sickness is caused by protozoan parasites of the genus Trypanosoma transmitted by tsetse flies (genus Glossina). Trypanosomes cause disease in both humans and animals and if left untreated, the disease is fatal [16]. The African trypanosomes pathogenic for humans belong to the species Trypanosoma brucei, which has two subspecies: T. b. gambiense, which causes the chronic form of the disease in central and west Africa; and T. b. rhodesiense, which causes more acute disease in East and Southern Africa (figure 1.1) [8, 11, 13, 47].
1.2
Epidemiology of African trypanosomiasis in Africa
Sleeping sickness is endemic in 37 countries of sub-Saharan Africa (figure 1.1), where there are suitable habitats for its vector, the tsetse fly. A small fraction of the patients in these countries are under surveillance with regular examination, have access to a health centre that can provide diagnostic facilities, or are protected by vector control interventions. The number of reported cases annually is over 10,000 [56]. Over 60 million people live in risk areas and since many of these have poor access to diagnostic and health care facilities, under-reporting is probably high [77, 79].
Chapter 1. Introduction 2
FIG. 1.1. Distribution of human African trypanosomiasis in sub-Saharan Africa. The black line divides the areas in which Trypanosoma bruce gambiense prevails and in which Trypanosoma brucei rhodesiense predominates [11].
Both human and animal African trypanosomiasis are a major cause of rural underdevel-opment in sub-Saharan Africa. Although there are a few cases reported in urban and peri-urban areas, it mainly affects poor and remote rural regions. Infections normally oc-cur in children and adults during activities such as farming, hunting, fishing, or washing clothes [11].
Three severe sleeping sickness epidemics took place in sub-Saharan Africa in the 20th
century. The first one took place between 1896 and 1906, and affected mainly Uganda and Congo killing an estimated 800,000 people. The second major epidemic which took place between 1920 and the late 1940s worried the colonial administrators and prompted them to
invest in vector and disease control. The disease was almost eradicated by the early 1960s (figure 1.2). By the mid 1960s most of the trypanosomiasis endemic countries had become independent and this led to the collapse of the surveillance and control activities due to political instability and economic ruin with a disastrous effect on the health services. This resulted in a decline in the number of screened individuals (figure 1.2), since most of the control programmes were stopped. The disease then re-emerged and reached its peak in the late 1990s. This marked the beginning of the third and most recent epidemic in the 20th century in sub- Saharan Africa, mainly affecting Angola, Congo, Southern Sudan and
the West Nile district of Uganda [11, 59].
0
1
2
3
4
5
6
1940 1950 1960 1970 1980 1990 2000
0
1
2
3
4
5
6
7
8
Number of cases (x10
4
)
Population screened (x10
6
)
Number of cases
Population screened
FIG. 1.2. Number of reported cases of African trypanosomiasis and population screened, 1939-2004. Data obtained from [55, 76]
Chapter 1. Introduction 4
1.3
Epidemiology of African trypanosomiasis in Uganda
In Uganda, African trypanosomiasis threatens the lives of more than 10 million people [58]. Both pathogens of human African trypanosomiasis are present in Uganda, with T. b. gambiense in the north-western and West Nile region, and T. b. rhodesiense in the Eastern region (figure 1.3). T. b. rhodesiense which was originally restricted to districts clustered around the north shore of Lake Victoria and the source of the Nille (Busoga region) has now spread to other districts including Tororo, Busia, Palisa and Mbale. Since 1980, the area affected by T. b. rhodesiense in Uganda has increased by a factor of 2.5 and the population at risk of getting the disease doubled [47]. This was observed through increased reporting of new cases of African trypanosomiasis in Uganda during 1977 to 1983. This increased reporting was attributed to the epidemic of T. b. rhodesiense sleeping sickness in Busoga (figure 1.4) [76]. Three major outbreaks have been recorded in Southeastern Uganda, with the latest epidemic starting in the 1980s [36, 72].
FIG. 1.3. The map of Uganda highlighting areas of T. b. gambiense and T. b rhodesiense transmission for the years 1985, 1995, 2000 and 2005 [26].
Livestock, in particular cattle is the most important reservoir of T. b. rhodesiense [26] in Eastern Uganda. Since the keeping of livestock is an economic and social activity in Eastern Uganda, the risk of transmission of the disease is anticipated to be high [9]. It is estimated
that cattle are kept in numbers of 1-5 cows per family either grazed at homestead or 50-100 animals grazed communally within or near the village borders [81]. Previous studies in Southeastern Uganda reported the prevalence of T. brucei species in the domestic cattle population to be 5%, and out of these, 23% being human infective T. b. rhodesiense. It has also been reported that a tsetse fly is five times more likely to pick up an infection from a cow than from a human [25].
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1940 1950 1960 1970 1980 1990 2000
Number of cases
FIG. 1.4. Number of reported cases of African trypanosomiasis in Uganda, 1939-2004. Data obtained from [55, 76]
Movement of infected animals between localities has been seen as a strong factor that influences the transmission of trypanosomiasis in many countries including Uganda. For example, animals were implicated in the transmission of T. b. rhodesiense disease during the 1940s epidemic in Busoga, Southeastern Uganda. Cattle restocking was also believed to have led to an outbreak of T. b. rhodesiense in 2000 in which 18% of cattle were found
Chapter 1. Introduction 6
to be carrying the human pathogen [8, 47]. In a study that was done in Tororo District, Uganda, it was established that over 50% of cattle traded in the market were originating from endemic sleeping sickness areas [25]. These studies indicate that there is a need for treating livestock that is to be moved from infected areas, either for restocking or selling, if the disease is to be put under control.
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Reported number of cases of T. b. rhodesiense in Africa, 1990−2009
Number of cases 0 500 1000 1500 2000
Number of cases in Uganda only Number of cases in Africa
FIG. 1.5. Number of reported cases of T. b. rhodesiense in Uganda compared with all the cases reported in Africa, 1990-2009. Data obtained from [55, 76]
1.4
Importance of domestic animals and wildlife in
the transmission of African trypanosomiasis
Trypanosomes are multi-host parasites capable of infecting a wide range of domestic and wildlife species, which constitute a reservoir for human infections. In domesticated animals clinical cases have been detected in cattle, water buffalo, sheep, goats, camels, horses, don-keys, alpacas, llamas, pigs, dogs, cats and other species. In wild animals clinical cases have been detected in bushbuck, buffalo, kudo,bushpig, duiker, giraffe, impala, lion, warthog,
waterbuck, zebra and other species [1, 71, 72]. In most parts of Africa, cattle are the main species affected, due to the tsetse feeding preferences and the fact that they can shield other domesticated animals such as goats and pigs from the effects of trypanosomiasis [61].
1.5
Parasite lifecycle and clinical aspects of infection
The parasite that causes human African trypanosomiasis is picked up from the blood of an infected human or domestic or wild animal by a tsetse fly while feeding and undergoes an essential maturation within the vector, resulting in the infectious stage in the salivary glands. Tsetse flies are more likely to pick an infection while having their first meal.
FIG. 1.6. The life cycle of African trypanosomiasis. Reproduced from [11]
Chapter 1. Introduction 8
to feed from an infected host on the first day of life. Other studies have also shown that infection rates in flies increase with increase in temperature [71]. The parasites are injected into a susceptible human or animal when the infectious vector is feeding. African trypanosomiasis appears in two stages, the first haemolymphatic stage and the second meningoencephalitic stage, which is characterised by the invasion of the central nervous system. For T. b. gambiense, infection is characterised by a chronic progressive course, with an estimated average duration of about 3 years, which is evenly divided between the first and second stages. T. b. rhodesiense disease is usually acute, and if left untreated death occurs within weeks or months. Figure 1.6 shows the life cycle of African trypanosomiasis in man and tsetse vectors. In man, the bloodstream forms a polymorphism with (A) dividing (black arrows) slender forms, (B) intermediate forms, and (C) stumpy forms. In the tsetse fly vector, the bloodstream forms transform to (D) dividing midgut forms, then to (E) the migrating epimastigote forms, which develop in the salivary glands to (F) the infective metacyclic forms, which are injected during the next blood meal into the mammalian host [11].
1.6
Control of African trypanosomiasis
Currently there are three principal control strategies for tsetse-transmitted trypanosomia-sis: trypanocidal drugs (chemotherapy and chemoprophylaxis), trypanotolerant cattle and tsetse control [40, 71]. Treatment of livestock in sub-Saharan Africa with trypanocidal drugs has been hindered by the problem of drug resistance [10, 33] and being expensive for many farmers. Treatment of human sleeping sickness is expensive, normally ranging from US$150 to US$800 per patient. Due to the toxicity of the drugs used for treating T. b. rhodesiense, about 5% of the patients die from the side effects of the treatment [80]. Moreover, many patients fail to report to health clinics or present with late stage symptoms [44], so prevention of the disease is a preferable option to reliance on curative treatment.
Based on these observations it has been suggested that controlling the disease by attacking its insect vectors, the tsetse flies (Glossina ssp) might be a preferable option [51]. Tsetse control methods include aerial and ground spraying, sterile insect technique, bait
tech-nology, and insecticide-treated cattle. Bait technology and insecticide-treated cattle do little damage to the environment and are very effective if applied properly in appropriate circumstances [33, 67].
Over the last two decades, there has been an increasing emphasis on getting farmers to control tsetse themselves, instead of relying on governments or donor organisations. The only feasible techniques that can be taken up by farmers as self-help schemes are bait methods, and the most cost-effective of these is the application of pyrethroids to cattle in tsetse infested areas. The original protocol for these applications involved the treatment of all the cattle, using the standard “whole-body” dose as recommended for tick control, and applied at intervals of about a month. To be effective, the technique must be applied over an area of at least several hundred square kilometers, necessitating participation by all livestock keepers over relatively large areas [10, 64, 67].
In areas where cattle provide the majority of tsetse blood meals, the use of insecticide-treated cattle provides one of the cheapest, and most effective, methods of controlling human African sleeping sickness. Research has shown, however, that only small areas of the animal need to be treated with the insecticide. Moreover, only the largest animals in any herd need to be treated since tsetse feed preferentially on the legs or belly of large or adult cattle. This leads to a large saving in insecticide costs and reduces the already small problems of insecticide pollution. Finally, this method offers the possibility of controlling ticks and tsetse problems simultaneously and thus, unlike other methods, can be integrated into the farmer’s tick control programme and the control programme can be undertaken by the farmers themselves [10, 31].
Insecticide-treated cattle have already been used in a number of countries in sub-Sharan Africa to control tsetse and trypanosomiasis. Some of these countries include Zambia [15], Zimbabwe [62, 64], Tanzania [28, 33], Ethiopia [7, 51], Burkina Faso [5, 6] and Uganda [39, 45]. The degrees of success differ in each of the countries where the control programme was carried out. As pointed by Hargrove et al, 2003 [32], the success of insecticide-treated cattle depends on the size and shape of the control areas, and the number and density of treated cattle. If the area treated is small, and is surrounded by a tsetse-infested area, invasion from the untreated area can re-infest all or much of the controlled region [63].
Chapter 1. Introduction 10
1.7
Cost-effectiveness analysis of disease control
strate-gies
Cost-effectiveness analysis in health care involves identification of all the relevant alterna-tive uses of resources (cost) and the evaluation of the expected health gains derived by putting those resources to use. The aim is to maximise the health benefits per dollar spent (or minimise the cost per unit of health benefit gained). Cost-effectiveness analysis is more useful in comparing broader sets of health policies or interventions to inform health-sector budget allocation decisions [20].
Cost-effectiveness analysis measures effects on both mortality (quantity of life) and mor-bidity (quality of life) [29, 42]. There are two methods of carrying out a cost-effectiveness analysis that combine the two effects. The quality-adjusted life years (QALY) is the first known method that was developed in the 1970’s. In this approach, the incremental effect of the control programme is compared to the status quo option in terms of the extension of life and reduction of time spent in disability. The QALY weights are coded on a scale of one for perfect health and zero for death [17, 52]. More details on the formulation of QALY calculations are given in [52].
The second method of cost-effectiveness analysis which was developed in the early 1990s that also combines morbidity and mortality effects is the Disability-Adjusted Life Year (DALY). This method was initiated by the World Health Organisation and is more com-monly used in the context of developing countries. In this method, DALY weights are coded on a scale of zero for perfect health and one for death, which is exactly the opposite of the QALY. The effect of interest in this case is the DALYs avoided, rather than QALYs gained. The DALY differs from the QALY in several aspects. Most importantly, the DALY incorporates an age-weighting function that assigns different weights to life years lived at different ages. The age-weights are lowest for the young and old, and peak at middle age when people are most productive. DALY weights have been constructed from a process of expert elicitation, while QALY weights are derived using a survey-based approach from the general population [17, 29, 42, 52].
In this thesis, the cost-effectiveness analysis is done using the DALY method and the details of DALY calculations are given in section 4.3.6.
1.8
Benefit-cost analysis of disease control
Cost-effectiveness analysis seeks to achieve a given health objective at lowest cost. Benefit-cost analysis, on the other hand, can be used to answer different types of questions of allocative efficiency. Unlike cost-effectiveness analysis, benefit-cost analysis is founded on a branch of economics known as welfare economics, which emphasises the public decisions that impact the economic interests of more than one person. A benefit-cost analysis requires monetising all health impacts by determining consumer’s willingness to trade income (or wealth) for the health improvements or a reduction in mortality risk. The proper decision rule is to select projects with the highest net benefits: total social benefits less total social costs. Benefit-cost analysis is intended to be a decision-making aid rather than the sole normative criterion for evaluating a policy or programme[17, 30]. More details on benefit-cost analysis calculations are given in section 5.2.2.
1.9
Integrated Control of Neglected Zoonotic diseases
(ICONZ) project
This project aims at improving human health and animal production in sub-Saharan Africa through the control of neglected zoonotic diseases in animals. With 21 partner institutions in Africa and Europe, and 8 case study areas in Africa, ICONZ is tackling eight ne-glected zoonoses at the moment. These zoonoses include: Anthrax, Bovine Tuberculosis, Brucellosis, Cysticercosis/Neurocysticercosis, Echinococcosis, Leishmaniasis, Rabies and Zoonotic Sleeping Sickness or Human African Trypanosomiasis (HAT).
In Uganda, ICONZ is involved in the control of Human African Trypanosomiasis (HAT) in the Southeastern region, where T. b. rhodesiense, the acute form of sleeping sickness is predominant. The case study for this project in Uganda is termed as “work-package” 8. The main objective of this study is to develop cost-effective disease control strategies for zoonotic trypanosomiasis and tick-borne animal diseases in Southeastern Uganda. This study which started in 2010 aims at collecting data for a series of interventions which will be used as input data for epidemiological models. The control activities being done include restricted application of insecticides (RAP) on cattle by treating 0, 25%, 50% and 100%
Chapter 1. Introduction 12
of village cattle at monthly intervals with and without trypanocidal intervention. The impact of the different intervention strategies will be monitored at monthly intervals and cost-effectiveness analysis will be done.
The work presented in this thesis gives theoretical models for the analysis of the data which is being collected from Tororo district, Southeastern Uganda.
1.10
Motivation
Despite the effort to control African trypanosomiasis and the vector responsible for the spread of Trypanosoma parasites, the disease still claims the lives of many people in Africa. The diagnosis of human African trypanosomiasis requires a high degree of training and expertise which is still lacking in sub-Saharan Africa. Its treatment requires a range of drugs that are limited, and only one of them is less than 40 years old. A number of problems are associated with the current therapies which include: resistance to, toxicity and lack of effective diagnostic drugs that are costly and cause adverse reactions [60]. More over, sleeping sickness affects the poorest and most disenfranchised rural communities with the least access to health care [25].
Donors and many African governments have reduced their commitment to tsetse control, leaving operations to local communities and other inexperienced agencies [67]. Affordable control measures are thus needed if epidemics are to be controlled. Treating cattle with insecticides is an increasingly important and cheap means of controlling tsetse flies and can be applied by farmers themselves. This control strategy is more likely to be sustained compared to other complex and expensive strategies such as aerial spraying and sterile insect technique [21, 64]. In this study, we intend to develop mathematical models that can be used to examine, assess and analyse the control of African trypanosomiasis in Southeastern Uganda using the cheapest and most affordable strategies of insecticide-treated cattle.
1.11
Objectives of the study
The aim of this research is to develop mathematical models that can be used to study the dynamics of the transmission and control of tsetse and trypanosomiasis through insecticide-treated cattle in Southeastern Uganda.
The specific objectives of this study are:
(i) To analyse the dynamics of the transmission of trypanosomiasis so as to find condi-tions necessary for the disease interruption, control and eradication.
(ii) To evaluate the relative contribution of treating cattle with insecticides through whole-body or restricted application on the control of tsetse and trypanosomiasis in Southeastern Uganda.
(iii) To evaluate the proportion of cattle needed to be treated with insecticides in order to significantly reduce the tsetse vector population and trypanosomiasis infection in each of the strategies in Southeastern Uganda.
(iv) To evaluate the cost-effectiveness and benefit-cost analysis of each of the control strategies.
1.12
Outline of this work
In Chapter 2 we give a review of the literature in mathematical modelling of trypanoso-miasis, with an emphasis on T. b. rhodesiense - the acute form of trypanosomiasis that is prevalent in East Africa. A review of studies done on the cost, benefit and effectiveness of trypanosomiasis control strategies is also given.
In Chapter 3 we give a model for the transmission of T. b. rhodesiense by tsetse vectors in human and cattle populations. Mathematical analysis and simulations of the model are given. Optimal control of T. b. rhodesiense through mass chemoprophylaxis and ITC is done.
Chapter 1. Introduction 14
In Chapter 4 we give a model for the transmission of T. b. rhodesiense by tsetse vectors in a multi-host population. Mathematical analysis of the model in absence of ITC is given. Numerical analysis of the model in the presence of ITC is also given. A cost-effectiveness analysis to evaluate the costs and effectiveness of controlling T. b. rhodesiense is done. In Chapter 5 we give a model for the control of T. b. rhodesiense through treatment of adult cattle only with insecticides. Numerical analysis of the model is given. A benefit-cost analysis to evaluate the benefits and benefit-costs of controlling T. b. rhodesiense is done. In Chapter 6, we give a detailed conclusion on our findings.
In the Appendix, we show how the tsetse recruitment rate was derived.
1.13
Publications
This dissertation was built around the following papers and presentations at conferences: Chapter 3:
• Modelling the control of Trypanosoma brucei rhodesiense through mass chemopro-phylaxis and insecticide-treated cattle, Damian Kajunguri, John W. Hargrove, Rachid Ouifki, Susan C. Welburn and Paul G. Coleman. To be submitted to PLOS Neglected Tropical Diseases.
The results of this paper were presented by D. Kajunguri at the DVTD/ITM Interna-tional Colloquium on Zoonoses and Neglected Infectious Diseases of Africa, November 1-4, 2011, Johannesburg, South Africa, in a talk entitled: A delay differential equa-tion model for the impact of mass chemoprophylaxis and insecticide-treated cattle on the control of T. b. rhodesiense.
Chapter 4:
• Modelling the control of tsetse and Trypanosoma brucie rhodesiense in a multi-host population through insecticide-treated cattle, Damian Kajunguri, John W. Hargrove, Rachid Ouifki, J.Y.T. Mugisha and Susan C. Welburn- in preparation
Part of this paper’s results were presented by D. Kajunguri at the the South African and American Mathematical Societies (SAMS/AMS) Congress, November 29 - De-cember 3, 2011, Port Elizabeth, South Africa in a talk entitled: Cost-effectiveness analysis of cheap and safe strategies for tsetse and sleeping sickness control.
• Cost-effectiveness analysis of tsetse and Trypanosoma brucei rhodesiense control through application of insecticides on cattle, Damian Kajunguri, John W. Hargrove, Rachid Ouifki, J.Y.T. Mugisha and Susan C. Welburn - in preparation.
The results of this paper were presented by D. Kajunguri at the Society for Math-ematical Biology 2012 Annual Meeting & Conference, July 25-28, 2012, Knoxville, Tennessee, USA.
Other publications. The following paper was influential in developing ideas related to the modelling of the control of tsetse and trypanosomiasis using trypanocides or insecticide-treated cattle.
• Modeling the control of trypanosomiasis using trypanocides or insecticide-treated livestock, John W. Hargrove, Rachid Ouifki, Damian Kajunguri, Glyn A. Vale and Stephen J. Torr, PLOS Neglected Tropical Diseases, 6, e1615.
Chapter 2
Literature review
A number of researchers have developed and analysed mathematical models that endeav-our to explain the transmission and control of trypanosomiasis. Research on the costs, benefits and effectiveness of various trypanosomiasis control measures has also been done. We present a literature review of some of the studies that have been done so far. We start by presenting modelling work that has been developed to explain the dynamics of trypanosomiasis transmission and control. We end with a review of studies that have been done to evaluate the costs, benefits and effectiveness of tsetse and trypanosomiasis control.
2.1
Modelling the transmission and control of
try-panosomiasis
One of the pioneering works in the modelling of trypanosomiasis was done by Rogers (1988) [50]. His model provides a mathematical framework to describe the vector-borne transmission of trypanosomes between multiple host species. The model assumed constant populations of tsetse flies and hosts and allowed for multiple tsetse, host and trypanosome species. The model was used to study trypanosomiasis in a typical West African village situation, with 300 humans, 50 domestic animals and 500 tsetse flies. The model predicted the equilibrium prevalence of T. vivax, T. congelense and T. brucei to be 47.0%, 45.8% and 28.7%, respectively, in the wild and domestic mammalian hosts, and 24.2%, 3.4% and 0.15% in the tsetse vectors. An equilibrium prevalence of 7.0% of the human-infective T.
brucei was also predicted in humans. It was demonstrated that the human infective T. brucei could not be maintained by humans hosts alone. This was due to the fact that the contribution to the basic reproduction number from human and animal hosts was 0.11 and 2.54, respectively. Seasonal changes that affect the fly mortality were shown to lead to an effect in both the future population size and infection rate. It was suggested that treating animal reservoirs will achieve a greater reduction in human sleeping sickness than direct treatment of humans alone.
A number of studies on the transmission of trypanosomiasis [16, 40, 72] have been based on the mathematical model developed by Rogers (1988) [50]. Model predictions for the trans-mission of T. b. brucei and T. b. rhodesiense in humans and cattle by one tsetse species, G. f. fuscipies show that T. b. rhodesiense would be 3 and 3.5-fold more prevalent than T. b. brucei in the cattle and vector populations, respectively [16]. It was also estimated that the cattle population accounts for approximately 92% of the total T. b. rhodesiense transmission potential, and human population contributing only 8% of the total basic re-production number. Adding the effects of both medical and veterinary interventions, it was shown that keeping about 86% of the cattle population effectively immune to infection interrupts the transmission of T. b. rhodesiense and protects the wider human population [72].
Reducing Rogers’ model to a one tsetse species, one trypanosome species (T. congelense) and one host species (cattle), McDermott and Coleman [40] used the model to evaluate the effects of four control measures on the transmission of trypanosomiasis in cattle. The control measures considered were curative drugs, vector control, use of trypanotolerant cattle and vaccination. Vaccination was taken to be a hopeful future option in the event that vaccines are developed. The results showed that the relative rankings of the effect of control strategies on reducing disease prevalence were: vector control, vaccination, and drug use, in that order. Trypanotolerance was assumed to decrease disease prevalence, but not to influence transmission.
Davis et al. (2011) [19] constructed models for the basic reproduction number of T. b. gambiense and T. b. rhodesiense, the causative agents of the West and East African human sleeping sickness, respectively. The models were used to carry out a global sensitivity analysis based on parameter ranges from the literature, field data and expertise out of
Chapter 2. Literature review 18
Uganda. For T. b. gambiense, the parameter for the proportion of blood meals taken from humans was found to be the most sensitive parameter to the basic reproduction number. The parameter for the proportion of tsetse flies refractory to infection was found to be the second ranked parameter for T. b. gambiense and the highest ranked for T. b. rhodesiense. The population parameters for tsetse species composition, survival and abundance were also ranked almost as high as the proportion refractory for T. b. rhodesiense. The results show the implications of nutritionally stressed tsetse that are more susceptible to trypanosomiasis infection and provides a broad support for control strategies that are aimed at increasing refractoriness in tsetse flies.
Recently, Hargrove et al. (2012) [31] generalised Rogers model and developed an R0
math-ematical model that allows tsetse to feed off multiple host species. They identified treat-ment coverages required to break transmission when host populations consisted of various proportions of wild and domestic mammals, and reptiles that support tsetse but do not harbour trypanosomes. The model was used to compare the control of trypanosomiasis through insecticide-treated cattle or treating cattle with trypanocides that protect against infection. The results show that in areas with few wild animals, where cattle provide most of the tsetse’s blood meals, treating cattle with insecticides could be a cheaper and more effective method for breaking transmission of trypanosomiasis. Assuming that tsetse feed only on cattle and humans, about 20% of the cattle need to be sprayed with insecticides to control the disease in humans, whereas 65% would be required to be treated with try-panocides to produce the same effect. Increasing the insecticide or trypanocide coverage to 55% or 100%, respectively, could lead to the control of T. congelense in cattle. T. vivax can only be controlled if 100% of the cattle population are kept on trypanocides. The results also showed that the presence of wild mammalian hosts lead to an increase in the coverage required and makes the control of T. congolense difficult. With insecticide-treated cattle, control of T. brucei and T. congelense is possible if proportions of non-human bloodmeals from cattle are more than 40% or 70%, respectively.
Mathematical models describing the dynamics of the spread of T. b. gambiense were presented in [2, 3, 4, 13]. The effects of tsetse immigration, vector control and detection (followed by treatment) of infected individuals were studied. The vector control strategy was shown to lead to the conrol of the Gambian sleeping sickness if the vector density is decreased by about fifty percent. Detection of sick individuals was found to be more
efficient if there is a long asymptomatic phase (first stage of Gambian sleeping sickness) characteristic of an endemic situation. It was also established that the persistence and/or extension of Gambian sleeping sickness foci could be due either to a continuous reinvasion of infected flies or to slow dynamics.
2.2
Comparative analysis of the costs, benefits and
effectiveness of tsetse and trypanosomiasis control
Shaw (1989) [54] made one of the first attempts to analyse the costs and benefits of different trypanosomiasis control strategies, which included vector control, human case finding and treatment. In the economic analysis, a benefit was defined as equivalent to one year’s infection avoided due to the control strategy for one person. Using a simple spreadsheet-based economic model, the relative economic performance of vector control versus human case finding and treatment in terms of the cost per benefit unit were compared. The analysis did not include domestic and wild animal populations. Though the results showed that the two control strategies are cost-effective, it was also pointed out that there is need to integrate economic and epidemiological models in order to evaluate control options. Wahab and Asuming-Bermpong (2007) [69] used a maximum Likelihood-Binary Logit model to estimate the cost of tsetse and treatment of trypanosomiasis and the benefits involved. The model was also used to estimate the extent to which socio-economic char-acteristics of farmers affect the use of tsetse control techniques in Ghana. Benefits were estimated using gains from effective disease control and revenue from increased cattle pro-duction. The cost of the disease was taken to be represented by the level of revenue forgone due to the disease (or revenue gained as a result of the control of the disease). The bene-fit/cost of the disease was determined by estimating the impact of trypanosomiasis, which consisted of estimates of the prevalence and incidence of infections and the effects of the disease on key livestock production parameters such as mortality, milk yield and draught power. The results showed that farmers will benefit if they invest in control and treatment of trypanosomiasis. The findings suggested that there is potential for farmer’s response and participation in tsetse control activities in Northern Ghana. It was recommended that more extension services be provided to livestock farmers to help them derive maximum
Chapter 2. Literature review 20
benefit from trypanosomiasis control practices.
A number of studies to estimate the cost-effectiveness of trypanosomiasis control were carried out based on the total costs of hospitalization and treatment for the disease. The health outcomes for different control options were rated against each other by looking at the DALYs averted. Studies are few and far between for human African trypanosomiasis, and have tended to focus on T. b. gambiense compared to T. b. rhodesiense. Shaw and Cattand [53] showed that above a prevalence of 2%, it is more cost-effective to screen and treat T. b. gambiense using mobile teams carrying out active surveillance. At lower prevalences, active screening may not be cost-effective in the short term. Lutumba et al. [37] estimated the cost-effectiveness of T. b. gambiense control programmes in terms of DALYs averted in Buma, Democratic Republic of Congo. In a population of 1, 300, active case finding control programme resulted in 1, 408 DALYs averted, for a cost of US$17 per DALY averted.
For T. b. rhodesiense, Fevre et al. [26] estimated the burden of the disease during an out-break in Serere, Uganda. Unique characteristics affecting the burden of T. b. rhodesiense such as age, severity, level of under-reporting and duration of hospitalisation were identified and put into consideration in quantifying the burden of Human African Trypanosomiasis (HAT). Early and late stage HAT morbidity were considered differently, and disability weightings were appropriately used for the T. b. rhodesiense form of HAT. The results showed that hospital-based interventions alone are cost-effective for HAT control in rural settings in Uganda, with a mean cost per DALY averted (for reported cases) of US$8.50. It was demonstrated that under-reporting accounts for 93% of the DALY estimate of T. b. rhodesiense.
2.3
Conclusion
An understanding of the pathogenesis and epidemiology of trypanosomiasis are essential in the development of mathematical models that can predict the dynamics for the trans-mission of the disease. In this Chapter we have reviewed some of the mathematical models used to describe the transmission dynamics of trypanosomiasis. But the model assump-tions, the number of parameters and equations used differ, and few of them have been
validated by experimental data. We acknowledge the limitations of data for parameter estimation. This has an impact of limiting the use of the mathematical models in real life scenarios. The following observations were made:
• It was noticed that most mathematical models describe the transmission dynamics of Trypanosoma brucei gambiense: little modelling work on Trypanosoma brucei rhodesiense has been done. It was also noticed that most studies on the economic burden of HAT were focused on T. b. gambiense rather than T. b. rhodesiense. • Most of the modelling work on trypanosomiasis is based on the basic reproduction
number analysis. This implies that the analysis is mainly focused on the transmission of trypanosomiasis, giving less or no information on the prevalence, incidence and progress of the disease over time.
• Mathematical models have been used to identify the role of the animal reservoir in the transmission and control of trypanosomiasis. Some of the findings indicate that interventions aimed at controlling the trypanosome parasite in the animal reservoir might lead to a greater reduction in human sleeping sickness compared to direct treatment of humans alone [16, 50].
Chapter 3
Modelling the control of T. b.
rhodesiense
through mass
chemoprophylaxis and
insecticide-treated cattle
3.1
Introduction
In this Chapter, we develop a simple mathematical model that describes the dynamics of the transmission of Trypanosoma brucei rhodesiense in humans, cattle and tsetse popula-tions. The model is an extension of the model developed by Rogers (1988) [50] describing the transmission of Trypanosoma brucei parasites by tsetse vector species between two host populations, to include the effects of veterinary and tsetse control interventions [72].
3.2
Model development and analysis
We consider the scenario in which a population of G. f. fuscipes tsetse flies, NV, transmits
T. b. rhodesiense parasites between populations of humans, NH, and cattle, NC. Tsetse
flies feed at a rate a per day so that each fly takes a new blood meal on average every 1adays. Vectors are assumed to feed on humans and cattle at random, but with a fixed preference taking a proportion fH of all meals from humans and the remainder fC = (1 − fH) of meals
from cattle. Thus, tsetse flies feed on humans at a rate aH = afH per day and on cattle at
a rate aC = afC per day, with a = aH + aC [72]. The human and cattle populations are
each divided into three classes, susceptible, infectious and recovered, whereas the tsetse population is divided into two classes, susceptible and infectious. We define time delays TH, TC and TV representing the incubation period in humans, cattle and tsetse vectors,
respectively. This means that infections are caused by infectious groups that were infected TH, TC and TV units of time earlier, respectively. A proportion, p and φC, of the cattle
population are assumed to be kept on insecticides and treated with trypanocides per day, respectively, to control T. b. rhodesiense. The tsetse mortality, µV(p), is taken to depend
on the proportion, p, of cattle that is kept on insecticides. Due to the short life span of tsetse flies and the effect of insecticide-treated cattle we assume that a proportion e−µV(p)TV of infected flies will survive the incubation period, TV [31, 50].
TABLE. 3.1. The model variables
Variable definition
SH Susceptible human population
IH Infectious human population
RH Recovered human population
SC Susceptible cattle population
PC Cattle population on treatment with trypanocides
IC Infectious cattle population
RC Recovered cattle
SV Susceptible tsetse population
IV Infectious tsetse population
We assume a constant tsetse birth rate BV. Only newly emerged flies (known as tenerals)
feeding in the first t days of adult life (where t < 1
a) are susceptible to T. b. rhodesiense infection [16, 31, 50]. The number of newly born flies that survive and have not fed in one unit of time (that is, between t − 1 and t) is given by
ΛV(p) = BV Z t t−1 e−(a+µV(p))(t−s) ds = BV (a + µV(p)) (1 − e−(a+µV(p)) ), (3.1)
where a is the tsetse feeding rate and µV(p) is the tsetse mortality which depends on the
proportion of cattle, p, kept on insecticides. The probability of a susceptible teneral fly acquiring infection is assumed to be α; for tsetse flies older than t = 1a days, the probability of acquiring infection is zero. It takes a total of TV days from acquiring trypanosomes for
Chapter 3. Modelling the control T. b. rhodesiense through mass chemoprophylaxis and
insecticide-treated cattle 24
TABLE. 3.2. Definitions of the parameters used in the model
Parameter Definition Parameter Definition
ΛH Recruitment rate for humans gC Recovery rate of infected cattle
fH Proportion of tsetse blood
meals from humans
µC Cattle natural mortality rate
aH Tsetse-human biting rate νC Rate of loss of immunity in recovered
cattle µH Human natural mortality
rate
TC Incubation period for cattle
gH Recovery rate of infected
hu-mans
φC Proportion of cattle treated with
try-panocides per day βH Probability of infected fly
bite producing an infection in humans
ǫ Protection provided to cattle by try-panocides towards T. b. rhodesiense infection (or efficacy of trypanocides) νH Rate of loss of immunity in
recovered humans
γC Rate of loss of chemoprophylactic
im-munity TH Incubation period for
hu-mans
p Proportion of cattle kept on insecti-cides
σH Mortality rate of infected
hu-mans
BV Tsetse birth rate
ΛC Recruitment rate for cattle ΛV(p) Recruitment rate for tsetse flies
fC Proportion of tsetse blood
meals from cattle
µV(p) Tsetse mortality rate
aC Tsetse-cattle biting rate TV Incubation period for tsetse flies
βC Probability of infected fly
bite producing an infection in cattle
α Probability of the first infected blood meal giving rise to infection in tsetse flies
σC Mortality rate of infected
cattle
a Tsetse feeding rate
the parasites to mature so that the fly becomes infectious. Infected tsetse flies are assumed to die of natural causes only.
We assume a constant recruitment rate ΛC and ΛH for cattle and humans, respectively,
which is through birth and emigration. Following the bite of an infectious tsetse fly, the probability of susceptible cattle and humans becoming infected is βC and βH, respectively.
Infectious cattle and humans are assumed to recover at a rate gC and gH, respectively.
Cattle are treated with trypanocidal drugs with efficacy of ǫ at a rate φC. Susceptible and
POPULATION HUMAN POPULATION TSETSE CATTLE POPULATION ΛC ΛV(p) ΛH SC PC IC RC SH IH RH SV IV γC φC gC gH νC νH µC µC µC µH µH σC σH µV(p) µV(p) λC (1 − ǫ)λC λV λH
FIG. 3.1. Flow diagram of the compartmental model of T. b. rhodesiense in cattle, human and tsetse populations. λC, λH, and λV are the forces of infection for cattle, humans and
tsetse vectors, respectively.
treatment. In the presence of treatment, it is implied that gC = gC′ + φC, where gC′ is the
self-recovery rate of infectious cattle. Recovered cattle and humans lose their immunity after a period of ν1
C and
1
νH, respectively, and become susceptible again. The flows between the different states (compartments) of the model are clearly shown in figure 3.1. The summary of model variables is given in Table 3.1. Table 3.2 shows the model parameters and their definitions.
3.2.1
Estimation of the tsetse mortality in the presence of
insecticide-treated cattle
Hargrove et al. (2012) [31] made the first attempt to estimate the tsetse mortality in the presence of ITC using the knowledge of probability theory and we adopt their method in this thesis. If the probability that a tsetse fly survives a feed on a host is q1 and the
Chapter 3. Modelling the control T. b. rhodesiense through mass chemoprophylaxis and
insecticide-treated cattle 26
probability of surviving a non-feeding day is q2, then a fly survives the feeding cycle of f
days, where f = 1
a with a probability
Sf = q1q2f. (3.2)
The daily mortality rate is calculated from µV = − ln(Sf)/f . Setting the parameters to
q1 = 0.96 and q2 = 0.98 as in [31], we obtain µV = 0.03, which is the natural mortality
that was assumed in [16, 50]. The survival probability in equation (3.2) can be re-written as
Sf = fCq1q2f + (1 − fC)q1q2f,
where fC is the probability that a fly feeds on cattle (which is equal to the proportion of
blood meals a tsetse fly takes from cattle) and 1 − fC is the probability that a fly feeds
on a non-cattle host. If the proportion of cattle treated with insecticides is p, then the probability of a fly surviving a feeding cycle is now given by
Sf = (1 − p)fCq1q2f + (1 − fC)q1qf2.
Thus, the tsetse mortality as a function of the proportion of cattle treated with insecticides, p, can be obtained as
µV(p) = − ln[(1 − p)fCq1q2f + (1 − fC)q1qf2]/f. (3.3)
of T. b. rhodesiense in cattle, human and tsetse populations. d dtSH = ΛH + νHRH −µHSH −λH(t − TH)SH(t − TH), (3.4) d dtIH = λH(t − TH)SH(t − TH) − (gH + σH)IH, (3.5) d dtRH = gHIH −(µH + νH)RH, (3.6) d dtSC = ΛC+ νCRC+ γCPC−(µC+ φC)SC−λC(t − TC)SC(t − TC), (3.7) d dtPC = φC(SC + IC + RC) − (µC+ γC)PC−(1 − ǫ)λC(t − TC)PC(t − TC), (3.8) d dtIC = λC(t − TC)[(1 − ǫ)PC(t − TC) + SC(t − TC)] − (φC + gC + σC)IC, (3.9) d dtRC = gCIC −(φC + µC+ νC)RC, (3.10) d dtSV = ΛV(p) − e −µV(p)TVλ V(t − TV)SV(t − TV) − µV(p)SV, (3.11) d dtIV = e −µV(p)TVλ V(t − TV)SV(t − TV) − µV(p)IV, (3.12) where λH(t) = aHβHIV(t)/NH(t) λC(t) = aCβCIV(t)/NC(t) and λV(t) = α[aHIH(t)/NH(t) + aCIC(t)/NC(t)].
The total population sizes can be determined by NH = SH + IH + RH, NC = SC + PC +
IC+ RC and NV = SV + IV or from the differential equations
d dtNH = ΛH −µHNH −(σH −µH)IH, (3.13) d dtNC = ΛC −µCNC −(σC−µC)IC, (3.14) d dtNV = ΛV(p) − µV(p)NV. (3.15)
By using standard results of well possedness, one can prove that the model (3.4-3.12) is well posed in the region
χ = (SH, IH, RH, SC, PC, IC, RC, SV, IV) ∈ R9+: NH ≤ ΛH µH , NC ≤ ΛC µC , NV ≤ ΛV(p) µV(p) .
Chapter 3. Modelling the control T. b. rhodesiense through mass chemoprophylaxis and
insecticide-treated cattle 28
Because of the presence of a delay term in the first equation of this model (and the models in Chapters 4 and 5), the classical proofs of its well possedness do not apply. Although, mathematically, this is an important point that should be further investigated. In this thesis, we focus more on the epidemiological properties of the model where this is not a serious issue.
Basic reproduction number, R0
The basic reproduction number, R0, is the average number of infectious cases an individual
would generate during his/her infectious period in a population that is wholly susceptible. In this case, the basic reproduction number is defined as the average number of T. b. rhodesiense cases that an infectious cattle or human or tsetse fly would generate in a totally susceptible population of humans, cattle and tsetse flies. If R0 > 1, then the
disease may emerge in one of the populations. However, if R0 < 1, then the disease-free
equilibrium is locally asymptotically stable [66]. Model system (3.4-3.12) has a disease-free equilibrium given by,
E0 = ΛH µH , 0, 0, ΛC(µC + γC) µC(µC + γC + φC) , φCΛC µC(µC+ γC + φC) , 0, 0,ΛV(p) µV(p) , 0 .
Using the next generation operator method described in [66], we obtain the matrices F (for the new infections) and V (for the transition terms) as
F = 0 0 aHβH 0 0 aCβC(µC+γC+(1−ǫ)φC) µC+γC+φC e−µV (p)TVαaHµHΛV(p) ΛHµV(p) e−µV (p)TVαaCµCΛV(p) ΛCµV(p) 0 and V = gH + σH 0 0 0 gC + σC 0 0 0 µV(p) ,
respectively. It follows that the basic reproduction number, denoted by R0, is given by
ρ(F V−1
) = q
R2
where ρ is the spectral radius (dominant eigenvalue in magnitude) of the next generation matrix F V−1 , R0H = s e−µV(p)TVΛ V(p)αa2HµHβH µ2 V(p)ΛH(gH + σH) (3.17) is the human-vector basic reproduction number and
R0C = s e−µV(p)TVΛV(p)αa2 CµCβC(1 − ǫπC) µ2 V(p)ΛC(gC + σC) , (3.18)
is the cattle-vector reproduction number, with πC = (φ φC
C+µC+γC) and (1 − ǫπC) > 0. The expressions of R0H and R0C are similar to the one obtained by Rogers (1988) [50] for
constant populations of humans, cattle and flies (except for the square roots, and the term (1 − ǫπC) which is due to treatment of cattle with trypanocides). The square root in the
expressions of R0H and R0C arises from the fact that two generations are required for an
infected vector or host to reproduce itself [66]. Note that in the derivation and analysis of R0, we are taking gC = g′C. A case where gC = gC′ + φC will be considered in numerical
simulations given in Section 3.4.
3.2.2
Effect of mass chemoprophylaxis and ITC on R0
From [66] we know that the disease-free equilibrium is locally asymptotically stable if and only if R0 < 1, that is ǫ ˜R20C(p) − ( ˜R02(p) − 1)φC > ( ˜R20(p) − 1)(µC + γC), (3.19) where ˜R2 0(p) = e−µV (p)TVΛV(p)α µ2 V(p) (AH + AC) and ˜R 2 0C(p) = e−µV (p)TVΛV(p)α µ2 V(p) AC, with AC = a2 CµCβC ΛC(gC+σC) and AH = a2 HµHβH
ΛH(gH+σH). Note that ˜R0(p) (resp. ˜R0C(p)) is the basic reproductive number, in the presence of ITC only, of the total population (resp. cattle population). Thus, we have the the following result
Proposition 1
1. If ˜R0(p) < 1, then there is no need for chemoprophylaxis as the disease will be
Chapter 3. Modelling the control T. b. rhodesiense through mass chemoprophylaxis and
insecticide-treated cattle 30
2. If ˜R0(p) > 1, then eradication is possible provided that
ǫ > ( ˜R 2 0(p) − 1) ˜ R2 0C(p) φC > ( ˜R20(p) − 1)(µC + γC) ǫ ˜R2 0C(p) − ( ˜R20(p) − 1) .
A simple differentiation of ˜R0(p) shows that it is a decreasing function of p. This implies
that
i). If ˜R0(0) < 1 (the basic reproductive number without ITC or prophylaxis), then
˜
R0(p) < 1 for each p ∈ [0, 1]
ii). If ˜R0(1) > 1, then ˜R0(p) > 1 for each p ∈ [0, 1]
iii). If ˜R0(0) < 1 < ˜R0(1), then there exists a unique p0 ∈ [0, 1] such that ˜R0(p0) = 1,
˜
R0(p) > 1 for p ∈ [0, p0) and ˜R0(p) < 1 for p ∈ (p0, 1].
3.2.3
The optimal control problem to minimise the cost of
con-trolling T. b. rhodesiense through chemoprophylaxis and
ITC
If we assume that the cost of treating one cow with trypanocides is C1 and the daily cost
of keeping one cow on insecticides is given by C2, then the optimal control problem can be
defined by
J(φC, p) = C1φC+ C2p, (3.20)
where J(φC, p) is the total daily cost of treating a proportion φC of cattle with trypanocides
and keeping a proportion p of cattle on insecticides. In proposition 1, we proved that T. b. rhodesiense control through ITC and chemoprophylaxis is possible provided that ǫ >
( ˜R2 0(p) − 1) ˜ R2 0C(p) and φC > ( ˜R2 0(p) − 1)(µC+ γC) ǫ ˜R2 0C(p) − ( ˜R20(p) − 1)
. Numerical simulations show that optimal control of T. b. rhodesiense for R0 < 1 and close to 1 can be achieved at a long period of
time (40 years for ITC only, and 95 years for ITC and chemoprophylaxis), which is very hard to be maintained. To solve this problem of disease control over a long period of time,
